Density of normal binary covering codes

Abstract

A binary code with covering radius $R$ is a subset $C$ of the hypercube $Q_n=\{0,1\}^n$ such that every $x\in Q_n$ is within Hamming distance $R$ of some codeword $c\in C$, where $R$ is as small as possible. For a fixed coordinate $i\in[n]$, define $C(b,i)$, for $b=0,1$, to be the set of codewords with a $b$ in the $i$th position. Then $C$ is normal if there exists an $i\in[n]$ such that for any $v\in Q_n$, the sum of the Hamming distances from $v$ to $C(0,i)$ and $C(1,i)$ is at most $2R+1$. We newly define what it means for an asymmetric covering code to be normal, and consider the worst case asymptotic densities $\nu^*(R)$ and $\nu^*_+(R)$ of constant radius $R$ symmetric and asymmetric normal covering codes, respectively. Using a probabilistic deletion method, and analysis adapted from previous work by Krivelevich, Sudakov, and Vu, we show that both are bounded above by $e(R\log R + \log R + \log\log R+4)$, giving evidence that minimum size constant radius covering codes could still be normal.